**The**

*n*th Term of Arithmetic Progression (Examples)

**Example 1:**

If the 20th term of an
arithmetic progression is 14 and the 40th term is –6,

Find

Find

**(a)**the first term and the common difference,

**(b)**the 10th term.

*Solution:***(a)**

*T*

_{20}= 14

*a*+ 19

*d*= 14 ----- (1) ←

**(**

*T*_{n}_{ }= a + (*n*– 1)*d*

*T*

_{40}= – 6

a
+ 39

*d*= – 6 ----- (2)
(2)
– (1),

20

*d =*– 20

*d =***– 1**

Substitute

*d =*– 1 into (1),*a*+ 19 (– 1) = 14

*a***= 33**

**(b)**

*T*

_{10}=

*a*+ 9

*d*

*T*

_{10}= 33 + 9 (– 1)

*T*

_{10}**= 24**

**Example 2:**

The 3rd term and the 7th term of an arithmetic progression are 20
and 12 respectively.

**(a)**Calculate the 20th term.

**(b)**Find the term whose value is – 34.

*Solution:***(a)**

*T*

_{3}= 20

*a*+ 2

*d*= 20 ----- (1) ←

**(**

*T*_{n}_{ }= a + (*n*– 1)*d*

*T*

_{7}= 12

a
+ 6

*d*= 12 ----- (2)
(2)
– (1),

4

*d =*– 8*d =*– 2

Substitute

*d =*– 2 into (1),*a*+ 2 (– 2) = 20

*a*= 24

*T*

_{20}=

*a*+ 19

*d*

*T*

_{20}= 24 + 19 (– 2)

*T*

_{20}**= –4**

**(b)**

*T*

_{n}= –34

a
+ (

*n*– 1)*d*= –34
24
+ (

*n*– 1) (–2) = –34
(

*n*– 1) (–2) = –58*n*– 1 = 29

*n***= 30**

**Example 3:**

The volume of water in a
tank is 75 litres on the first day. Subsequently, 15 litres of water is added
to the tank everyday.

Calculate the volume, in
litres, of water in the tank at the end of the 12th day.

*Solution:*
Volume of water on the
first day = 75

*l*
Volume of water on the
second day = 75 + 15 = 90

*l*
Volume of water on the
third day = 90 + 15 = 105

*l*
75, 90, 105, …..

AP,

*a*= 75,*d*= 90 – 75 = 15
Volume
of water on the 12

^{th}day,*T*

_{12}=

*a*+ 11

*d*

*T*

_{12}= 75 + 11 (15)

*T*

_{12}**= 240**

*l***Example 4:**

The first three terms of
an arithmetic progression are 72, 65 and 58.

The

*n*th term of this progression is negative.
Find the least value of

*n*.

*Solution:*
72,
65, 58

AP,

*a*= 72,*d*= 65 – 72 = –7
The
nth term is negative,

*T*

_{n }< 0

*a*+ (

*n*– 1)

*d*< 0

72
+ (

*n*– 1)*(–7) < 0*
(

*n*– 1)*(–7) < –72**n*– 1 > –72

**/**–7

*n*– 1 > 10.28

*n*> 11.28

*n*must be integer,

*n*= 12, 13, 14, ….

**Therefore, the least value of**

*n*= 12.